"He" refers to God. "This dream" was the gift of Christ. The gift was to reconcile us and bring us back to God. (Mark 10:14-16 When Jesus saw this, he was indignant. He said to them, "Let the little children come to me, and do not hinder them, for the kingdom of God belongs to such as these. I tell you the truth, anyone who will not receive the kingdom of God like a little child will never enter it." And he took the children in his arms, put his hands on them and blessed them.)
Ok, but then what does "This dream He had each child still knows" refer to? The Bible never refers to Jesus having a dream, so if it is in the context of the story of the birth of Jesus, then "he" could be Joseph, to whom an angel appeared in a dream both before (see Mt 1:20-24) and after (see Mt 2:13,19) Jesus was born. But does "each child" still know the content of that dream? Does each Christian adult even know it?